Abstract
I discuss Yablo’s approach to truthmaker semantics and compare it with my own, with special focus on the idea of a proposition being true of or being restricted to some subject-matter, the idea of propositional containment, and the development of an ‘incremental’ semantics for the conditional. I conclude with some remarks on the relationship between truth-maker approach and the standard possible worlds approach to semantics.
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Notes
I am grateful for the comments of the participants in a seminar on Yablo’s book, held in the Fall of 2014 at NYU, and for the comments of the participants in a workshop on the book, held in the Summer of 2015 at the University of Hamburg, under the auspices of an Annalieser Maier Award from the Humboldt Foundation. I am especially grateful to Steve Yablo, both for his comments and for his inspiring work.
For some reason, which I do not fully understand, my paper was not published along with Yablo’s response (Yablo 2018). I only discovered this fact much later and, as a consequence, there is now a significant lapse in time between Yablo’s response and the paper to which it was a response. In re-reading my paper, I realize that I should have stated my objection to an initial objection to an intensional conception of states (which Yablo rightly criticizes) somewhat differently and that I should also have framed the issues concerning the development of a general theory of the conditional somewhat differently. However, in the interests of presenting the paper to which Yablo actually responded, I have kept the paper exactly as it was but for this addition to the footnote, the update to some references and a few minor changes.
Let us grant [1]+. Then [1]− can be derived from the additional assumptions that sm−(A) = sm+(¬A) and that |¬A| = <P′, P> when |A| = <P, P′>. For then sm−(A) = sm+(¬A) = σ(|¬A|+) = σ(P′) = σ(|A|−).
However, I should perhaps mention that Yablo and I do not altogether see eye to eye on the first question, although this may be more a matter of disagreement in emphasis than principle. I much prefer the recursive approach to truthmaking to the reductive approach (in contrast to §§4.3, 4.6) and I am inclined to regard the truthmaker content of a sentence as principally a semantic rather than a pragmatic matter (in contrast to §4.7). This calls for detailed discussion but let me simply mention how we might deal with one of Yablo’s concerns, relating to the conditional (p. 60). Yablo wants p → p ∧ q to be true because of the verifier q for q, whereas on the standard recursive account p → p ∧ q is equivalent to ¬p ∨ (p ∧ q) and is either verified by a falsifier for p or a verifier for p ∧ q. But, on the recursive account, we do not need to accept this equivalence and, as we shall see (Sect. 8), there is a recursive account which appears to deliver the results he wants.
The identification is implicit in his remark that ‘No new machinery is required…S’s reasons, or ways, for being true are just additional [truth-conditional] propositions’ (p. 2) and his ‘cellular’ treatment of subject matter (p. 45), under which any state can be regarded as a cell, i.e. a non-empty set of worlds, within some subject matter. However, the reader should also take note of Yablo’s attempt to accommodate the impossible in the appendix to chapter 5 (pp. 92–94).
The worlds may, of course, be further identified—as models or valuations or the like (p. 61).
I do not like the term ‘truth-conditional’, since I think the truth-conditions of a sentence are more plausibly taken to be its exact verifiers. Calling the intensional content ‘truth-conditional’ is to give it a false sheen. But the terminology is, at this stage, probably too well established to be relinquished.
For present purposes, I am ignoring Yablo’s intensional conception of states and I am also presupposing that the sentence A is true or false in each possible world and that the state space contains the full panoply of possible worlds (is a ‘W-space’ in a terminology that I shall later introduce). One can also set up the correspondence without the use of this latter assumption but, in this case, the intensional content X of A must be identified with a suitable set of states from the worlds which, if they were to exist, would verify the sentence.
Two states are said to be compatible when their fusion is a possible state.
I have taken the principles to follow to be about the subject matter of propositions but there are, of course, analogous principles (and points to be made) about the subject matter of sentences.
I have here, for simplicity, assumed a certain semantics for disjunction, under which the positive content of A ∨ B will be the union of the positive contents of A and B. There are other semantic clauses that might be given for disjunction, and for the other Boolean connectives. Most of what I say will apply, mutatis mutandis, to these other cases, although I have not aimed for full generality.
This is not the only way of defining the partial (see Fine 2018b). But any reasonable definition will yield similar results.
If P contains some verifier other than □, then P′ can be obtained from P° by adding the part □ and so the subject matters of P, P° and P′ are the same. If P only contains □, then P° = ∅ and P′ = {□} and so we only need to be assured that the null proposition ∅ and the null state proposition {□} are subject-matter-identical. But this will presumably be so, given that the null state □ is “contentless”. Finally, if P is empty then so are P° = ∅ and P′.
Yablo also requires each verifier of Q to contain a verifier of P (p. 30). But this would then prevent the subject matter of P being, in general, part of the subject matter of P ∨ Q and so it seems to me that this further requirement should be dropped.
Under an obvious representation, let the worlds be {p, q}, {p, ¬q}, {¬p, q} and {¬p, ¬q} and the cells be {{p, q}, {¬p, ¬q}} and {{¬p, q}, {p, ¬q}}. Then no state is common to the worlds of either cell.
In different contexts, different states may count as the states, or candidate verifiers. It is any interesting question whether every possible intensional proposition corresponds to a state in some or other context. But even if the answer is ‘yes’ (which I very much doubt), the point remains that, relative to any given context, there will usually be some restriction on what the candidate verifiers and hence on what the possible subject matters might be.
He also wishes to allow a subject matter to be defined on a proper subset of the worlds (p. 30), though this will not concern us so much in what follows.
See especially fn 27 (p. 37), where he writes “I have decided for practical reasons to stick with divisions, leaving covers to footnotes, but “really” the whole thing should be redone with them” and he makes clear that we would then lose the one–one correspondence with similarity relations.
For R must relate any two distinct worlds and hence should be a relation for which {w1, w2, w3} is a maximal set of similar worlds and hence a cell. The example is from Hazen and Humberstone (2004) (Example 1.1), which is a general study of the correspondence between divisions and similarity relations.
Curiously, a counter-example of this sort does not arise within a “canonical” state space, in which all states are formed from a class of independent atomic states.
These two forms of agreement come to the same thing for Lewis: two worlds will agree on a cell in a partition just in case they agree on every cell in the partition. Yablo is aware of these two ways of defining agreement (or difference) (fn. 40, p. 40) but goes astray, from our own point of view, in opting for the former.
Note that the intermediate closure P* and the maximal closure P** will result in the same notion of equivalence (or agreement) between worlds. Thus even though it is possible to recover P** from the resulting equivalence relation, it is not possible, in general, to recover P*.
The interpretation of the text is complicated by the fact that his three formulations of containment on pp. 45–46 [(11), (12), and (13)] are not, in fact, equivalent. For given the definition of subject matter inclusion at (5), the second clause of (11) will require that every verifier of A imply a verifier of B, where there is no such requirement in (12) and (13). I assume that (12) and (13) give Yablo’s true intent and that, as before, the definition of subject matter inclusion at (5) should be appropriately weakened.
I should also note that the discussion of relevant containment on p. 59 is unduly specific in a number of ways. For there is no need (i) to tie it to a recursive conception of truthmakers, (ii) to identify states with sets of literals, or (iii) to suppose that the truthmakers of atomic sentences are themselves atomic. Here, as elsewhere, we should take ourselves to be operating within an arbitrary state space.
I do not discuss the role of overall subject matter in characterizing containment here, though it is relevant to the conception of containment in Parry (1933), under which P ∧ Q, for example, will contain ¬P ∨ Q.
Strictly speaking, an overall subject matter is never a Lewisian subject matter, since the former is a pair of divisions, which is not of the right set-theoretic type to be a partition. But the overall subject matter {{V}, {W − V}} might be taken to correspond to the bipartite partition {V, W − V}.
The resulting impossible states must be finely individuated and, in contrast to the possible states, cannot be identified with sets of worlds. In particular, the fusion of possible states, when the fusion is impossible, cannot be identified with the intersection of the corresponding cells.
The proof rests upon the not implausible assumption that s conform to the condition that if p is incompatible with q ⨅ s then p ⨅ s is incompatible with q ⨅ s. Let us also note that, even though the mereological union p ⨆ q can be identified, in the intensional framework, with the set-theoretic intersection of p and q, the mereological intersection p ⨅ q cannot be properly identified with the set-theoretic union of p and q, since the union, from an intuitive point of view, may not be a state. Thus reference to the underlying state space is required to make sense of the notion. I am not aware that Yablo ever makes reference to the mereological intersection of two states but, as we see, it does important work in justifying some of his definitions.
In specifying the intensional proposition associated with A/m (clauses (1) and (3), p. 53), Yablo seems to take for granted that the proposition will be bivalent.
We need to suppose that the subject matter s satisfies the condition from footnote 24. Yablo mentions another difficulty (fn. 15, p. 32), which is that ‘there is not always such a thing as a part of A about m’ for ‘it will have… to be included in A’s subject matter a’ and ‘connect up somehow with m’, which will be impossible ‘if m and a are unrelated’. But I would have thought that the part of A about m will be the part about the common part of the subject matter of A and m, which will be the “null” subject matter when m and a are unrelated.
The quotations are from an earlier unpublished version of the paper. I have not been able to find them in the published paper.
A minimal truthmaker for a sentence is a truthmaker no proper part of which is a truthmaker for the sentence (p. 61).
For the purposes of the comparison, I have reversed the usual ordering in a Heyting algebra.
We might follow Yablo (p. 185) and take the conditional to be true (false) simpliciter only when it is true (false) without being false (true), although there are other means by which truth-value gluts might be avoided.
I do not know if it was the requirement that truthmakers should be incomparable that led Yablo to reject □ as a truthmaker. But, as we have seen, this is not a requirement that should have been made in the first place.
Perhaps somewhat oddly, Yablo in his previous discussion of the truthmakers for a universal generalization ∀xFx in §4.4 does not take the truthmakers to be relative to a world which contains certain individuals but takes them instead to presuppose that these are the individuals.
I myself am somewhat unhappy with truthmakers carrying presuppositions in this way. Truthmakers are in the world; and it seems to me that presupposition properly belongs to language, not to the world.
We might define w | = A to hold when w loosely verifies A or when it inexactly verifies A (the two are equivalent in the case of worlds).
These are clauses I myself have adopted in some unpublished work on the semantics for imperatives. There is a question, which I shall not consider, of how clauses like the ones above can be extended to conditionals
in which A or C may also contain a conditional.Stated more rigorously, the result is that s ⊩w A in model over S iff w|s ⊩ A in the corresponding model over SW × S.
Yablo also aims for a unified account of conditionals in his paper, but along very different lines.
Made at the Hamburger workshop on Yablo’s book.
Another notable example is the Amsterdam school of inquisitive semantics.
in which A or C may also contain a conditional.References
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Fine, K. Yablo on subject-matter. Philos Stud 177, 129–171 (2020). https://doi.org/10.1007/s11098-018-1183-7
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DOI: https://doi.org/10.1007/s11098-018-1183-7